The progress in computer technology during the last 10-15 years has
enabled the performance of ever more precise quantum mechanical
calculations related to structure and interactions of chemical
compounds. However, the qualitative models relating electronic structure
to molecular geometry have not progressed at the same pace. There is a
continuing need in chemistry for simple concepts and qualitatively clear
pictures that are also quantitatively comparable to ab initio quantum
chemical calculations. Topological methods and, more specifically, graph
theory as a fixed-point topology, provide in principle a chance to fill
this gap. With its more than 100 years of applications to chemistry,
graph theory has proven to be of vital importance as the most natural
language of chemistry. The explosive development of chemical graph
theory during the last 20 years has increasingly overlapped with quantum
chemistry. Besides contributing to the solution of various problems in
theoretical chemistry, this development indicates that topology is an
underlying principle that explains the success of quantum mechanics and
goes beyond it, thus promising to bear more fruit in the future.